The Independent-Samples t Test

Chapter 11

Quick Test Reminder

> One person = Z score> One sample with population standard

deviation = Z test> One sample no population standard

deviation = single t-test> One sample test twice = paired samples

t

Independent Samples t-Test

> Used to compare two means in a between-groups design (i.e., each participant is in only one condition)• Remember that dependent t (paired

samples) is a repeated measures or within-groups design

Did you notice?

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Between groups design

> In between groups, your sets of participant’s scores (i.e. group 1 versus group 2) have to be independent• Remember independence is the

assumption that my scores are completely unrelated to your scores

Quick Distributions Reminder

• Z = Distribution of scores• Z = distribution of means (for samples)• t = distribution of means (for samples with

estimated standard deviation)• t = distribution of differences between

paired scores (for paired samples with estimated standard deviation)

• t = distribution of differences between means (for two groups independent t)

Distribution of Differences Between Means

Hypothesis Tests & Distributions

Let’s talk about Standard Deviation

Test Standard Deviation Standard deviation of distribution of …(standard error)

Z σ (population) σM

Single t s (sample) sM

Paired t s (sample on difference scores)

sM

Independent t s group 1s group 2spooled

sdifference

Let’s talk about Standard Deviation

Variance = same for all tests, but paired t is on difference scores

Standard error = same for paired and single t

Take the square root for standard deviation of these

Let’s talk about Standard Deviation

222Y

total

YX

total

Xpooled s

df

dfs

df

dfs

222

YX MMdifference sss

Variance = same for all tests, but paired t is on difference scores

This section is for independent t only

Take the square root for standard deviation of these

Let’s talk about test statisticsTest type Formula

Z M – μM

σM

Single t M – μM

sM

Paired t M sM

Independent t M – Msdifference

Additional Formulae

difference

YXYX

s

MMt

)()(

difference

YX

s

MMt

Let’s talk about dfTest type df

Single sample N – 1

Paired samples t N – 1

Independent t N – 1 + N – 1

Assumptions

Assumption Solution

Normal distribution N >=30

DV is scale Nothing – do nonparametrics

Random selection (sampling) Random assignment to group

Steps for Calculating Independent Sample t Tests

> Step 1: Identify the populations, distribution, and assumptions.

> Step 2: State the null and research hypotheses.

> Step 3: Determine the characteristics of the comparison distribution.

> Step 4: Determine critical values, or cutoffs.

> Step 5: Calculate the test statistic.> Step 6: Make a decision.

SPSSYou’ll need two columns:1) Group membership column

(independent variable)1) Remember use value

labels! Helps with reading the output

2) Dependent variable column

SPSS

Analyze > compare means > Independent Samples T-Test

SPSS

IV goes in grouping variable, DV into Test variable

SPSS

Hit define groups. Put in the numbers that you used to define groups

SPSS

Hit options – if you need a 99% CI, you can change it here.

SPSS

N values for each groupM values for each groups values for each groupsM values for each group (not the standard error you need)(Confidence intervals here)

SPSS

t = step 5df = step ¾Mdifference = M – MStd. Error = SdifferenceCI = CI around Mdifference (what the book suggests)

> t(df) = tcalc, p < .05 • Use p > .05 if there is no difference

between means• Use p < .05 if there is a difference between

means

> t(7) = -2.44, p < .05

Reporting the Statistics

Beyond Hypothesis Testing

> Just like z tests, single-sample t tests, and paired-samples t tests, we can calculated confidence intervals and effect size for independent-samples t tests

Steps for Calculating CIs

> The suggestion for CI for independent t is to calculate the CI around the mean difference (M – M).• This calculation will tell you if you should reject

the null.• Does not match what people normally do

(which is calculate each M CI separately).

Effect Size

> Used to supplement hypothesis testing> Cohen’s d:

pooled

YXYX

s

MMd

)()(